Theoretical framework for energy flux analysis of channels under drag control

A framework for analyzing energy flux in turbulent channel flows is proposed which enables quantification of the drag reduction efficacy by different control methods. In contrast to the FIK [Fukagata, Iwamoto, and Kasagi, Phys. Fluids 14, L73 (2002)] and the RD [Renard and Deck, J. Fluid Mech. 790, 339 (2016)] identities, this framework expresses the skin friction coefficient in terms of the nondimensionalized dissipation rate and the work done by external excitation. We extend the energy-box analysis of Gatti et al. [J. Fluid Mech. 857, 345 (2018)] through a triple decomposition of energy flux and show how mean (${\epsilon }_M$), coherent (${\epsilon }_C$), and random turbulent (${\epsilon }_R$) dissipations contribute differently to the drag reduction and the net power saving. Three control methods, including our recently developed spanwise opposed wall-jet forcing (SOJF), the spanwise wall oscillation (SWO), and the opposed wall blowing/suction (OBS) controls, are compared at ${\text{Re}}_{\tau }\approx 200$ via direct numerical simulations (DNS). While all methods yield comparable drag reductions ($\sim 20\%$), OBS yields the maximum net power saving, followed by SOJF, and then SWO. Specifically, for SOJF control, ${\epsilon }_C$ (induced by the large-scale swirls) is much smaller than ${\epsilon }_R$ (induced mainly by the small-scale near-wall vortices) and ${\epsilon }_M$ (due to the spanwise vorticity sheet ${\mathrm{\Omega }}_z$). In contrast, for SWO control, ${\epsilon }_C$ (caused by the wall oscillation-induced ${\mathrm{\Omega }}_x$ vortex sheet)—much larger than that of SOJF—is comparable to ${\epsilon }_R$ and ${\epsilon }_M$. For OBS control, ${\epsilon }_R$ is notably suppressed without any introduction of ${\epsilon }_C$ as the energy is injected through the random velocity field. Diagnoses performed at a higher ${\text{Re}}_{\tau }$ (i.e., 2000) for SWO shows that random turbulent dissipation ${\epsilon }_R$ predominates due to the increasing near-wall vortical structures—hence their suppression should be the target for drag control at high ${\text{Re}}_{\tau }$. The analysis also suggests a promising hybrid drag control strategy by incorporating both the random (OBS) and coherent (SOJF or SWO) controls together, an issue for future exploration.

Figure 1

Sketch of energy flux box in channel by (a) Reynolds decomposition and (b) triple decomposition. Note that all the symbols have been time and volume averaged (the brackets $\langle \rangle$ have been omitted). Abbreviations MKE, TKE, CKE, and RKE represent mean, turbulent, coherent, and random kinetic energies, respectively. Drag reduction is indicated by the saving of pumping power $P$, while the net power saving by the additional subtraction of control power $W$ ($=W_c+W_R$).

Figure 2

Energy budget at ${\text{Re}}_{\tau }=200$ under OBS control. (a) Mean kinetic energy and (b) turbulent fluctuation kinetic energy. Here, we take dissipation as negative to balance other terms.

Figure 3

Energy flux in a channel at ${\text{Re}}_{\tau }=200$ (a) without control and (b) under OBS control. All the numbers are normalized by the pumping power of flow without control. Note that the drag reduction is 23% (indicated by the pumping power saving) and the net power saving is 22% (after a subtraction of control power).

Figure 4

Energy budget at ${\text{Re}}_{\tau }=200$ for (a) coherent kinetic energy and (b) random fluctuation kinetic energy under SWO control. The inset of (a) is a zoom-in plot of the coherent kinetic energy budget.

Figure 5

Energy flux in a channel at ${\text{Re}}_{\tau }=200$ under SWO control with (a) $A^{+}=8$ and (b) $A^{+}=12$.

Figure 6

Energy budget at ${\text{Re}}_{\tau }=200$ for (a) coherent kinetic energy and (b) random fluctuation kinetic energy under SOJF control.

Figure 7

Energy flux in turbulent channel under SOJF for ${\text{Re}}_{\tau }=200$.

Figure 8

Bar chart of drag reduction rate by ${\mathcal{N}}_M$, suppression of dissipation of mean flow, ${\mathcal{N}}_R$, suppression of random turbulent dissipation, ${\mathcal{N}}_C$, coherent dissipation induced by control, and ${\mathcal{N}}_{\text{in}}$, power input by control, for (a) OBS, (b) SWO, and (c) SOJF control; all at ${\text{Re}}_{\tau }=200$. Note that OBS control involves zero coherent dissipation as the energy is injected to the flow through random turbulence fluctuation.

Figure 9

Profiles of mean dissipation, random turbulent dissipation, and coherent dissipation (all normalized by the pumping power $P_0$) for channels at ${\text{Re}}_{\tau }\approx 200$ with OBS control (dashed line), SWO control (dotted line), SOJF control (dashed-dotted line), and no control (solid line).

Figure 10

Distribution of random turbulent dissipation under SOJF control at ${\text{Re}}_{\tau }=180$. Top left: Isosurface of random turbulent dissipation (${\epsilon }_R=1.5\times {10}^{-3}$) colored by $u$. Top right: Isosurface of ${\lambda }_2$ structure (${\lambda }_2=-0.35$) colored by $u$. Bottom left and right: Contours of ${\epsilon }_R$ (red lines) and ${\lambda }_2$ (blue lines) at the cross planes $x/L_x=1/3$ and $5/8$, respectively. The two cross planes are marked in both top panels.

Figure 11

Energy flux for (a) without control and (b) under SWO control at ${\text{Re}}_{\tau }=2000$.